A Complete Guide for Tensors computations using Physics
This is an old request, a complete guide for using Physics to perform tensor computations. This is it. Feedback on how to improve this presentation is welcome. The Guide, shown below with Setions closed, is linked at the end of this post as a Maple worksheet that allows for reproducing its contents, including all the hyperlinks. Most of the illustrative computations shown are reproducible in Maple 2018.2.1, and a significant part also in previous releases, but to reproduce everything shown below you need to have the Maplesoft Physics Updates version 283 or higher installed.
Physics is a package developed by Maplesoft, an integral part of the Maple system. In addition to its commands for Quantum Mechanics, Classical Field Theory and General Relativity, Physics includes 5 other subpackages, three of them also related to General Relativity: Tetrads , ThreePlusOne and NumericalRelativity , (work in progress, currently named Cactus) plus one to compute with Vectors , and another related to the Standard Model (also work in progress)..
This material is organized as follows. Section I is complete regarding the functionality for computing with tensors provided with the Physics package, in Classical and Quantum Mechanics (so including Euclidean spaces), Electrodynamics and Special Relativity. The material of section I is also relevant for working with tensors in General Relativity, for which section II is all devoted to curved spacetimes. )(The subsection on the NewmanPenrose formalism needs to be filled with more material and a new section devoted to the EnergyMomentum tensor is appropriate. I will complete these two things as time permits.) Section III is about transformations of coordinates, relevant in general.
For a alphabetical list of the Physics commands with a brief oneline description and a link to the corresponding help page see Physics: Brief description of each command .

I. Spacetime and tensors in Physics


This section contains all what is necessary for working with tensors in Classical and Quantum Mechanics, Electrodynamics and Special Relativity. This material is also relevant for computing with tensors in General Relativity, for which there is a dedicated Section II. Curved spacetimes .

Tensors, their definition, symmetries and operations


Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of General Relativity. In addition, one of the strengths of Physics is that you can define tensors, in natural ways, by indicating a matrix or array with its components, or indicating any generic tensorial expression involving other tensors.
In Maple, tensor indices are letters, as when computing with paper and pencil, lowercase or upper case, latin or greek, entered using indexation, as in , and are displayed as subscripts as in . Contravariant indices are entered preceding the letter with ~, as in , and are displayed as superscripts as in . You can work with two or more kinds of indices at the same time, e.g., spacetime and space indices.
To input greek letters, you can spell them, as in mu for , or simpler: use the shortcuts for entering Greek characters . Rightclick your input and choose Convert To → 2D Math input to give to your input spelled tensorial expression a textbook high quality typesetting.
Not every indexed object or function is, however, automatically a tensor. You first need to define it as such using the Define command. You can do that in two ways:
1. 
Passing the tensor being defined, say , possibly indicating symmetries and/or antisymmetries for its indices.

2. 
Passing a tensorial equation where the lefthand side is the tensor being defined as in 1. and the righthand side is a tensorial expression  or an Array or Matrix  such that the components of the tensor being defined are equal to the components of the tensorial expression.

After defining a tensor  say or  you can perform the following operations on algebraic expressions involving them
• 
Simplify products using Einstein's sum rule for repeated indices.

• 
SumOverRepeatedIndices of the tensorial expression.

• 
Use TensorArray to compute the expression's components

If you define a tensor using a tensorial equation, in addition to the items above you can:
• 
Get each tensor component by indexing, say as in or

• 
Get all the covariant and contravariant components by respectively using the shortcut notation and .

• 
Use any of the special indexing keywords valid for the predefined tensors of Physics; they are: definition, nonzero, and in the case of tensors of 2 indices also trace, and determinant.

• 
No need to specify the tensor dependency for differentiation purposes  it is inferred automatically from its definition.

• 
Redefine any particular tensor component using Library:RedefineTensorComponent

• 
Minimizing the number of independent tensor components using Library:MinimizeTensorComponent

• 
Compute the number of independent tensor components  relevant for tensors with several indices and different symmetries  using Library:NumberOfTensorComponents .

The first two sections illustrate these two ways of defining a tensor and the features described. The next sections present the existing functionality of the Physics package to compute with tensors.



II. Curved spacetimes


Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of general relativity, respectively entered and displayed as: Einstein[mu,nu] = , Ricci[mu,nu] = , Riemann[alpha, beta, mu, nu] = , Weyl[alpha, beta, mu, nu], = , and the Christoffel symbols Christoffel[alpha, mu, nu] = and Christoffel[~alpha, mu, nu] = respectively of first and second kinds mentioned in the previous section. The Tetrads and ThreePlusOne subpackages have other predefined related tensors. This section is thus all about computing with tensors in General Relativity.

